# Solution: 2014-08 Two positive integers

Let $$a$$, $$b$$ be distinct positive integers. Prove that there exists a prime $$p$$ such that when dividing both $$a$$ and $$b$$ by $$p$$, the remainder of $$a$$ is less than the remainder of $$b$$.

The best solution was submitted by 이종원 (2014학번). Congratulations!

Alternative solutions were submitted by 황성호 (+3), 정성진(+2), 박훈민 (+2). There were a few incorrect submissions (KSJ, JKJ, KDS, AHS, KKS, PKH).

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For integer $$n \geq 1$$, define
$a_n = \sum_{k=0}^{\infty} \frac{k^n}{k!}, \quad b_n = \sum_{k=0}^{\infty} (-1)^k \frac{k^n}{k!}.$
Prove that $$a_n b_n$$ is an integer.